**Control-Oriented Reduced-Order Models for Lithium-Metal Batteries**

22nd Advanced Automotive Battery Conference– San Diego, CA – December 5-8, 2022

**Authors:** Aloisio Kawakita de Souza, Wesley Hileman, M. Scott Trimboli, Gregory L. Plett

Department of Electrical and Computer Engineering, University of Colorado Colorado Springs

### Background

- Lithium-metal batteries (LMB) hold promise as successors to lithium-ion batteries (LIB) due to its high energy-density.
- LMB have metallic lithium anodes which introduce complications to modelling their long-term behaviour. Particularly, a dead-lithium layer grows over time, and this must be described in any battery-management-system (BMS) model to enable accurate estimates of state-of-charge, state-of-health, and power limits.

### Objectives

Develop two physics-based control-oriented models of LMBs for the design of advanced BMS algorithms:

- A reduced-order discrete-time model generated via a subspace-based method and the linearized transfer functions derived from the fundamental equations.
- A simplified model based on polynomial functions that compute the solid diffusion in the positive electrode and the electrolyte dynamics across the cell.

### Pseudo Two-Dimensional Model

- The pseudo two-dimensional (P2D) model adopted from [1] describes the processes for the electrolyte phase in the 𝑥 dimension across the cell sandwich. While in the positive solid electrode, the porous structure is represented by spherical particles with radial dimension 𝑟 at each channel location in 𝑥.

- Moreover, Xu et al. [1] models the Li metal negative electrode as an electrode surface with a high electrical conductivity and includes a static description of the dead-lithium layer at a particular state-of-age of a LMB cell. It does not specify how the dead-lithium layer evolves as a cell ages.
- This model is the foundation of our work, but it is too computationally complex to be used directly in real-time BMS algorithms.

### Enhanced Single Particle Model

- The single particle model (SPM) is based mainly on the two following assumptions [2]:
- [A1] The positive electrode is assumed to be a single spherical particle whose surface area is equivalent to the active area of the porous electrode.
- [A2] The total moles of lithium in the electrolyte and in the solid phase are both conserved.

- The assumptions above implicate that:
- (a) The lithium concentration in the solid phase does not depend on spatial distance 𝑥 across the electrode.
- (b) The intercalation reaction flux is assumed to be proportional to the applied current.

- The electrolyte dynamics is incorporated to the SPM to improve voltage prediction at high C-rate applications.
- Polynomial functions are used to approximate the solution of the solid diffusion and electrolyte concentration resulting in a linear state-space system [3].

### Physics-Based Reduced-Order Model

- The physics-based reduced-order model (ROM) is obtained by a transfer-function based approach that generates linear state-space (SS) models at different SOC and temperature setpoints.

- The realization algorithms (xRAs) optimizes the error between the approximate model and the discrete-time frequency response of the electrochemical variables for a specific 𝑥 location of interest.

- The linear SS models are blended together at every time step using a bilinear interpolation scheme between the four closest models to the present SOC and temperature.

### Simulation Results

- Simulation results for the ROM and SPM are compared against a virtual cell implemented in COMSOL using the cell parameters from [1].
**Case 1: Constant current discharge**- Cell is discharged from 90% to 5% SOC at 0.5, 1, 2 and 3 C-rates.

**Case 2: Dynamics driving profile**- Cell is discharged from initial SOC of 80 % using an Urban Dynometer Driving Schedule (UDDS)

### Highlights

- Physics-based control-oriented models have been developed for LMBs based on previous work on LIBs.
- The simulation results for ROM and SPM are compared against a virtual cell implemented in COMSOL via FOM using the cell parameters from [1].
- ROM has shown superior performance over the SPM for constant current discharge profiles above 2 C-rate.
- Both ROM and SPM demonstrated accurate voltage predictions for dynamics driving profiles
- Future work will include temperature dynamics, aging and experimental results.

### References

- S. Xu, K.-H. Chen, N. Dasgupta, J. Siegel, and A. Stefanopoulou, “Evolution of dead lithium growth in lithium metal batteries: Experimentally validated model of the apparent capacity loss,” Journal of The Electrochemical Society, vol. 166, pp. A3456–A3463, 01 2019.
- S. J. Moura, F. B. Argomedo, R. Klein, A. Mirtabatabaei, and M. Krstic, “Battery State Estimation for a Single Particle Model with Electrolyte Dynamics,” IEEE Trans Contr Syst Technol, vol. 25, no. 2, pp. 453–468, Mar.
- Rahimian, S. K., Rayman, S., & White, R. E. “Extension of Physics-Based Single Particle Model for Higher Charge-Discharge Rates. Journal of Power Sources”, 224, 180 – 194.
- Lu, D. “Identifying Physical Model Parameter Values for Lithium-Ion Cells”. Ph.D. Thesis, University of Colorado Colorado Springs, Colorado Springs, CO, USA, 2022.

**Control-Oriented Reduced-Order Models for Lithium-Metal Batteries**

22nd Advanced Automotive Battery Conference– San Diego, CA – December 5-8, 2022

**Authors:** Aloisio Kawakita de Souza, Wesley Hileman, M. Scott Trimboli, Gregory L. Plett

Department of Electrical and Computer Engineering, University of Colorado Colorado Springs

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